Abstract
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation as t approaches the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of the initial data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berryman J.G., Holland C.J.: Stability of the separable solution for fast diffusion. Arch. Rational Mech. Anal. 74, 379–388 (1980)
Blanchet A., Bonforte M., Dolbeault J., Grillo G., Vázquez J.L.: Asymptotics of the fast diffusion equation via entropy estimates. Arch. Rational Mech. Anal. 191, 347–385 (2009)
Bonforte M., Dolbeault J., Grillo G., Vázquez J.L.: Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proc. Natl. Acad. Sci. 107, 16459–16464 (2010)
Bonforte M., Grillo G., Vázquez J.L.: Special fast diffusion with slow asymptotics Entropy method and flow on a Riemannian manifold. Arch. Rational Mech. Anal. 196, 631–680 (2010)
Bonforte, M., Grillo, G., Vázquez, J.L.: Behaviour near extinction for the fast diffusion equation on bounded domains. J. Math. Pures Appl. doi:10.1016/j.matpur.2011.03.002
Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., Unterreiter A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133, 1–82 (2001)
Carrillo J.A., Lederman C., Markowich P.A., Toscani G.: Poincaré inequalities for linearizations of very fast diffusion equations. Nonlinearity 15, 565–580 (2002)
Carrillo J.A., Vázquez J.L.: Fine asymptotics for fast diffusion equations. Commun. Partial Differ. Equ. 28, 1023–1056 (2003)
Daskalopoulos P., Sesum N.: On the extinction profile of solutions to fast diffusion. J. Reine Angew. Math. 622, 95–119 (2008)
Denzler J., McCann R.J.: Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology. Arch. Rational Mech. Anal. 175, 301–342 (2005)
Dolbeault J., Toscani G.: Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic Relat. Models 4, 701–716 (2011)
Fila M., Vázquez J.L., Winkler M.: A continuum of extinction rates for the fast diffusion equation. Commun. Pure Appl. Anal. 10, 1129–1147 (2011)
Fila M., Winkler M., Yanagida E.: Convergencerate for a parabolic equation with supercritical nonlinearity. J. Dyn. Differ. Equ. 17, 249–269 (2005)
Fila M., Winkler M., Yanagida E.: Slow convergence to zero for a parabolic equation with a supercritical nonlinearity. Math. Ann. 340, 477–496 (2008)
Fila M., Winkler M., Yanagida E.: Convergence to self-similar solutions for a semilinear parabolic equation. Discrete Contin. Dyn. Syst. 21, 703–716 (2008)
Vázquez J.L.: Asymptotic behaviour for the porous medium equation posed in the whole space. J. Evol. Equ. 3, 67–118 (2003)
Vázquez, J. L.: Smoothing and decay estimates for nonlinear diffusion equations. Oxford Lecture Notes in Maths. and its Applications, vol. 33. Oxford University Press, Oxford, 2006
Vázquez J. L.: The porous medium equation Mathematical theory Oxford Mathematical Monographs. Oxford University Press, Oxford (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Šverák
Rights and permissions
About this article
Cite this article
Fila, M., Vázquez, J.L., Winkler, M. et al. Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation. Arch Rational Mech Anal 204, 599–625 (2012). https://doi.org/10.1007/s00205-011-0486-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0486-z